Helicity of neutrinos

All neutrinos are left handed and all anti neutrinos are right handed.so,it should be lorentz invariant and travel at speed of light. if it travels at c,then it is massless. but, neutrino oscillation requires mass? why there is contradiction?

All neutrinos are left handed and all anti neutrinos are right handed.so,it should be lorentz invariant and travel at speed of light. if it travels at c,then it is massless. but, neutrino oscillation requires mass? why there is contradiction?

You don't even really need helicity to get a "contradiction". Neutrino oscillation can happen only of neutrinos of different flavors have different masses. If a neutrino is created by the Sun in an electronic flavor, it has a given energy and a given momentum. How can it be measured on Earth with a different mass if energy and momentum are conserved ?

I mean to say that the issue of symmetries in neutrino oscillation can not be thought of academically without proper treatment of (elementary) quantum effects. The answer is indeed what Clem points out : the problem is academic because the symmetry violations are so tiny, they merely stem from the Heisenberg's inequalities. I don't mean to say it is trivial either, such a proper treatment is rather tedious.

All neutrinos are left handed and all anti neutrinos are right handed.so,it should be lorentz invariant and travel at speed of light. if it travels at c,then it is massless. but, neutrino oscillation requires mass? why there is contradiction?

Well, you hit the nail on the head. All neutrinos are left handed (and anti-neutrinos right handed) only in the approximation mass=0. Now that we know they have mass, this rule simply isn't true any more (that is, it's possible to define a reference frame where neutrinos are right-handed).

The usual terminology as I understand it is to call particles that enter the weak interaction with V-A (The sign is different in some formalisms.) are called "left handed" so that all leptons are left handed, even if they do not have pure alignment.

I think there is a confusion between helicity and chirality. The weak interaction couples to the left-handed chiral component of the neutrino. If the neutrino were massless, that would also be the left-handed polarization state of the neutrino.

Since the neutrino has mass, one can boost to a frame where the polarization becomes right-handed, but there is still a left-handed chiral component, so it still feels the weak force.